CLEVELAND — Scrambling among Indiana politicians has reached the point where Republican Party insiders are convinced that presumptive presidential nominee Donald Trump will pick Indiana Gov. Mike Pence as his running mate.

Constitutional lawyer James Bopp, an Indiana delegate to the Republican National Convention who is close the governor, told The Washington Times that Indiana House Speaker Brian C. Bosma, 58, a conservative Republican, had sought advice from him on running for governor.

“He wanted my counsel on what he needed to do to set himself up to run for governor, because he expects Pence to step down as governor in order to be Trump’s running mate,” Mr. Bopp said in an interview.

The Trump election team boosted the Pence speculation Sunday by suddenly adding a campaign rally in Indianapolis to a fundraiser planned for Tuesday featuring Mr. Trump and Mr. Pence.

Several Republicans close to the campaign and to the governor have told The Times over the last 24 hours that they are now convinced it’ll be Mr. Pence.

Mr. Bopp, also a member of the convention’s rules committee, told The Times that the “rally made it a 95 percent probability it’s Pence.”

Indiana Republican Party Chairman Jeff Cardwell said that Sunday’s sudden addition of a Trump rally after the fundraising event was a complete change from Mr. Trump’s original schedule, which had called for a quick appearance at the fundraiser and equally quick exit from Indiana.

Mr. Pence, who is little known nationally but highly admired in conservative circles, also made a telling private call to Mr. Cardwell, according to a Republican close to both men.

In the call the governor told Mr. Cardwell to delay his planned Tuesday departure to Cleveland for a Republican National Committee meeting, saying Mr. Cardwell needed to be sure to attend an Indianapolis fundraiser featuring Mr. Trump and Mr. Pence.

The Trump-Pence fundraiser already was a big deal, with tickets are going from $2,700 to $250,000. But its scheduled date falling so close to the July 18-21 Republican National Convention here was being interpreted by some political observers as ideal for a possible VP announcement by the party’s presumptive presidential nominee.

But Mr. Cardwell also said the request to delay his departure for Cleveland came from the RNC, not the governor, and had nothing to do with “Trump announcing Pence as his running mate.”

Until now, former House Speaker Newt Gingrich was the most probable vice presidential pick in most press accounts and in the minds of many GOP activists. But Republicans close to the action began to have doubts when Mr. Trump, at a rally with Mr. Gingrich last week, assured the crowd that the Georgian would definitely serve in some capacity in a Trump administration.

Those assurances seemed designed to gently let down loyalists Mr. Gingrich, 73, has accumulated over his 36 years in politics and public life, during which he is credited with planning the GOP’s stunning 1994 takeover of the U.S. House after 40 years in the minority.

Mr. Gingrich has traveled with Mr. Trump on campaign stops and has won kudos from even Trump skeptics for explaining the hows and whys of the Trump phenomenon at this point in American political history better than anyone else, including Mr. Trump himself.

Mr. Gingrich has been a national figure in politics virtually from the day he first won election to the U.S. House in 1977 and has made political enemies for his brashness and unorthodox ideas.

According to campaign insiders, Mr. Trump genuinely likes Mr. Gingrich but some in the Trump family, especially his adult children, aren’t great Gingrich fans.